Let $V$ and $W$ be finite dimensional irreducible representations of a Lie Algebra or a Lie Superalgebra. If $V$ is one dimensional, is $V\otimes W$ necessarily irreducible?
I know this to be true for $\mathfrak{gl}(n)$ and the superalgebra $\mathfrak{gl}(m|n)$ and I wonder if this is true in general. Thanks.
A 1-dimensional represenation $V$ is always invertible, in the sense that there is another representation $V^{-1}$ such that $V\otimes V^{-1}$ and $V^{-1}\otimes V$ are both isomorphic to the trivial representation. Indeed, you can tale $V^{-1}$ equal to the dual (that is, contragredient) repreentation of $V$.
It follows from that that if $V\otimes W$ is isomorphic to a dirct sum $U\oplus U'$, with $U$ and $U'$ nonzero, then $V^{-1}\otimes(V\otimes W)$ is isomorphic both to $V^{-1}\otimes U\oplus V^{-1}\otimes U'$ and to $W$. This is impossible if $W$ is irreducible.